Hydrodynamic modeling and experimental characterization of the plasmonic and thermoelectric
terahertz response of field-effect transistors with integrated
broadband antennas in AlGaN/GaN HEMTs and
CVD-grown graphene
Dissertation
zur Erlangung des Doktorgrades der Naturwissenschaften
vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe-Universität
in Frankfurt am Main
von
Maris Bauer
aus Frankfurt am Main
Frankfurt am Main 2017 (D30)
Johann Wolfgang Goethe-Universität als Dissertation angenommen.
Dekan: Prof. Dr. Michael Lang Gutachter:
Prof. Dr. Hartmut Roskos Prof. Dr.-Ing. Viktor Krozer
Prof. Dr.-Ing. Peter Haring Bolívar Datum der Disputation: 18.10.2018
“Wer versteht schon alles, was er so schreibt?”
Für meinen Vater
Terahertz (THz) physics are an emerging field of research dealing with electromagnetic radiation in the far-infrared to microwave region. The development of innovative technologies for the generation and detection of THz radiation has only in the recent past led to a tremendous rise of both fundamental research as well as investigation of possible fields of application for THz radiation. The most prominent reason has long been the scarce accessibility of the THz region of the electromagnetic spectrum - commonly loosely located between 0.1 and 30 THz - to broad research, and it was mostly limited to astronomy and high energy physics facilities. Over the recent years, numerous novel concepts on both the source and detector side have been proposed and successfully implemented to overcome this so-called THz gap. New technology has become available and paved the way for wide-spread experimental laboratory work and accompanying theoretical investigations. First application studies have emerged and in some cases even commercial development of the field of THz physics is on the rise.
Despite these enormous progresses, a continuing demand for more efficient THz detectors still impels current technological research. Relatively low source powers are often a major limiting factor and the request for new detection concepts, their understanding and implementation, as well as the optimization on a device basis has been and still remains in place. One of these concepts is the use of field-effect transistors (FETs) high above their conventional cut-off frequencies as electronic THz detectors. The concept has been proposed in a number of theoretical publications by M. Dyakonov and M. Shur in the early 1990’s, who pioneered to show that under certain boundary conditions, non-linear collective excitations of the charge carrier system of a two-dimensional electron gas (2DEG) by incident THz radiation can exhibit rectifying behaviour - a detection principle, which has become known as plasma wave or plasmonic mixing. Up until this day, the concept has been successfully implemented in many device realizations - most advanced in established silicon CMOS technology - and stands on the edge of becoming commercially available on a large scale.
The main direction of the work presented in this thesis was the modeling and experimental characterization of antenna-coupled FETs for THz detection - termed TeraFETs in this and the author’s previous works - which have been implemented in different material systems. The materials presented in this thesis are AlGaN/GaN HEMTs and graphene FETs. In a number of scientific collaborations, TeraFETs were designed based on a hydrodynamic transport model, fabricated in the respective materials, and characterized mainly in the lower THz frequency region from 0.2 to 1.2 THz.
The theoretical description of the plasma wave mixing mechanism in TeraFETs, as initiated by Dyakonov and Shur, was based on a fluid-dynamic transport model for charge carriers in the transistor channel. The THz radiation induces propagating charge density oscillations (plasma waves) in the 2DEG, which via non-linear self- mixing cause rectification of the incident THz signals. Over the course of this work, it became evident in the on-going detector characterization experiments that this original theoretical model of the detection process widely applied in the respective literature does not suffice to describe some of the experimental findings in TeraFET detection signals. Thorough measurements showed signal contributions, which are identified in this work to be of thermoelectric origin arising from an inherent asym- metric local heating of charge carriers in the devices. Depending on the material, these contributions constituted a mere side effect to plasmonic detection (AlGaN/GaN) or even reached a comparable magnitude (graphene FETs). To include these effects in the detector model, the original reduced fluid-dynamic description was extended to a hydrodynamic transport model. The model yields at the current stage a reasonable qualitative agreement to the measured THz detection signals.
This thesis presents the formulation of a hydrodynamic charge carrier transport model and its specific implementation in a circuit simulation tool. A second modeling aspect is that the transport equations cover only the intrinsic plasmonic detection process in the active gated part of the TeraFET’s transistor channel. In order to model and simulate the behavior of real devices, extrinsic detector parts such as ungated channel regions, parasitic resistances and capacitances, integrated antenna impedance, and others must be considered. The implemented detector model allows to simulate THz detection in real devices with the above influences included.
Besides presentation of the detector model, experimental THz characterization of the fabricated TeraFETs is presented in this work. Careful device design yielded record detection performance for detectors in both investigated materials. The respective results are shown and the experimental observations of the thermoelectric effect in TeraFETs are compared to modeling results. It is the goal of this work to provide a framework for further theoretical and experimental studies of the plasmonic and thermoelectric effect in TeraFETs, which could eventually lead to a new type of THz detectors particularly exploiting the thermoelectric effect to enhance the sensitivity of today’s plasmonic TeraFETs.
Die Terahertz-(THz)-Physik ist ein aufstrebendes Forschungsfeld, welches sich mit elektromagnetischer Strahlung vom Ferninfraroten bis hin zum Mikrowellenbereich befasst. Die Entwicklung innovativer Technologien zur Erzeugung und Detektion von THz-Strahlung führte erst in der jüngeren Vergangenheit zu einem gewaltigen Anstieg von Grundlagenforschung sowie der Auslotung möglicher Anwendungsfelder von THz Strahlung. Der vorrangige Grund war die lange Zeit spärliche Zugänglichkeit des THz- Bereichs - üblicherweise lose veranlagt zwischen 0.1 und 30 THz - für breite Forschung und der Bereich war hauptsächlich beschränkt auf Einrichtungen der Astronomie und der Hochenergiephysik. In den letzten Jahren wurden zahlreiche neuartige Konzepte sowohl auf der Seite von Quellen als auch Detektoren vorgeschlagen und erfolgreich implementiert um diese sogenannte THz-Lücke zu überwinden. Neue Technologie wurde zugänglich und ebnete den Weg für breit angelegte experimentelle Laborarbeit und begleitende theoretische Untersuchungen. Erste Anwendungsstudien wurden durchgeführt und in einigen Fällen ist sogar die kommerzielle Erschließung der THz Physik auf dem Vormarsch.
Trotz dieses enormen Fortschritts wird die technologische Forschung noch immer durch die anhaltende Nachfrage nach effizienteren THz-Detektoren angetrieben. Oft- mals sind geringe Ausgangsleistungen von Quellen der vorherrschende, limitierende Faktor und die Anforderung an neue Detektionskonzepte, deren Verständnis und Implementierung, sowie die Optimierung auf Bauelementebene, waren und sind stets vorhanden. Eines dieser Konzepte ist die Anwendung von Feldeffekttransistoren (FETs) weit über deren herkömmlichen Cut-off Frequenzen als elektronische THz-
Detektoren. Das Konzept wurde in einer Reihe von theoretischen Beiträgen von M.
Dyakonov und M. Shur anfang der neunziger Jahre vorgeschlagen, als diese erstmals zeigten, dass unter bestimmten Randbedingungen nichtlineare kollektive Anregungen des Ladungsträgersystems eines zweidimensionalen Elektronengases (2DEG) durch die einfallende THz-Strahlung gleichrichtend wirken können - das Detektionsprinzip wurde bekannt unter dem Namen Plasmawellen- oder plasmonisches Mischen. Bis heute wurde das Konzept in zahlreichen konkreten Bauelementen erfolgreich imple- mentiert - am weitesten fortgeschritten in etablierter Silizium CMOS Technologie - und steht vor dem Sprung in großem Maße kommerziell verfügbar zu werden.
Die zentrale Ausrichtung der in dieser Arbeit vorgestellten Aktivitäten war die Modellierung und experimentelle Charakterisierung antennengekopplter FETs für die THz Detektion - in dieser und in früheren Arbeiten des Autors TeraFETs genannt - welche in verschiedenen Materialsystemen implementiert wurden. Die in dieser Arbeit vorgestellten Materialien sin AlGaN/GaN HEMTs und Graphene-FETs. In einer Reihe wissenschaftlicher Zusammenarbeiten wurden TeraFETs auf der Basis eines
hydrodynamischen Transportmodells entworfen, in den entsprechenden Materialien hergestellt und hauptsächlich im unteren THz-Frequenzbereich von 0.2 bis 1.2 THz charakterisiert.
Die theoretische Beschreibung des Mechanismus des Mischens mit Plasmawellen in TeraFETs, wie ursprünglich von Dyakonov und Shur angestoßen, basierte auf einem fluiddynamischen Transportmodel für Ladungsträger innerhalb des Transistorkanals.
Die THz Strahlung induziert laufende Ladungsträgerdichtewellen (Plasmawellen) innerhalb des 2DEGs, welche durch nicht-lineares Selbstmischen eine nichtlineare Gle- ichrichtung des einfallenden THz-Signals bewirken. Im Verlaufe dieser Arbeit zeigte sich im Zuge der Detektionsexperimente, dass dieses weitverbreitete, ursprüngliche, theoretische Model des Detektionsmechanismus nicht aussreicht einige der experi- mentellen Befunde in den Detektionssignalen zu erklären. In gründlichen Messungen zeigten sich Signalbeiträge, welche in dieser Arbeit als thermoelektrisch identifiziert werden, ausgelöst durch ein inhärentes, lokales Anheizen von Ladungsträgern in dem Bauelement. In Abhängigkeit des Materials stellten diese Beiträge lediglich einen Nebeneffekt zur plasmonischen Detektion dar (AlGaN/GaN) oder erreichten eine ver- gleichbare Stärke (Graphen FETs). Um diese Effekte in das Detektormodell zu inter- grieren wurde das ursprüngliche fluiddynamische Modell zu einem hydrodynamischen Transportmodell erweitert. Das Modell erzielt in der jetzigen Form eine vernünftige, qualitative Übereinstimmung mit den gemessenen THz-Detektionssignalen.
In dieser Arbeit wird die Formulierung eines hydrodynamischen Ladungsträger- transportmodells und dessen spezifische Implementierung in einem Schaltkreis-Simu- lationstool dargestellt. Ein weiterer Aspekt der Modellierung ist, dass die Transport- gleichungen nur den intrinsischen plasmonischen Detektionsprozess in dem aktiven gegateten Teil des Transistorkanals abdecken. Um realistische Bauelemente zu mod- elieren und zu simulieren müssen extrinsische Elemente, wie z.B. ungegatete Teile des Kanals, parasitäre Widerstände und Kapazitäten, die Impedanz der integrierten Antenne u.a. berücksichtigt werden. Das implementierte Detektormodell erlaubt eine Simulation der THz-Detektion in realen Bauelementen unter Berücksichtigung dieser Einflüsse.
Neben der Darstellung des Detektormodells wird die experimentelle Charakte- risierung der hergestellten TeraFETs präsentiert. Ein gründliches Design der De- tektoren führte zu Rekordwerten der Detektionsleistung in beiden dargestellten Materialien. Die entsprechenden Messergebnisse werden gezeigt und die experi- mentelle Beobachtung des thermoelektrischen Effekts in TeraFETs verglichen mit Ergebnissen der Modellierung. Es ist das Ziel dieser Arbeit ein Rahmenwerk für weitere theoretische und experimentelle Studien des plasmonischen Mischens und des thermoelektrischen Effekts in TeraFETs zu liefern, welche letztendlich zu einer neuen Art von THz Detektoren führen könnten, welche im Speziellen den thermoelektrischen Effekt ausnutzen um die Sensitivität heutiger plasmonischer TeraFETs zu erhöhen.
Abstract iv
Zusammenfassung vi
1 Introduction 1
1.1 Detection mechanisms in TeraFETs . . . 1
1.2 Hydrodynamic transport model and circuit implementation . . . 3
1.3 Fabrication and experimental characterization of TeraFETs . . . 5
1.4 Structure of this thesis . . . 5
2 Plasma wave mixing model 7 2.1 Foundation of the concept of plasma wave mixing . . . 7
2.2 One-dimensional treatment of charge carrier transport . . . 9
2.3 Intrinsic channel impedance and THz response . . . 11
2.3.1 Channel impedance and carrier velocity . . . 11
2.3.2 THz response . . . 14
2.4 Detection regimes . . . 16
2.4.1 Classical resistive mixing . . . 17
2.4.2 Distributed resistive mixing . . . 18
2.4.3 Plasmonic mixing . . . 19
2.5 Asymmetric boundary conditions . . . 21
2.6 Transmission line equivalent circuit description . . . 23
3 Hydrodynamic model 26 3.1 The Boltzmann transport equation . . . 26
3.1.1 The distribution function . . . 28
3.1.2 The method of moments . . . 29
3.1.3 Balance equations . . . 30
3.2 Transport models . . . 31
3.2.1 Charge continuity equation . . . 31
3.2.2 The drift-diffusion model . . . 32
3.2.3 Hot carriers . . . 33
3.2.4 The hydrodynamic transport model . . . 35
3.3 Comparison of the Dyakonov-Shur and the hydrodynamic transport model . . . 36
CONTENTS
4 Circuit model implementation 39
4.1 The intrinsic distributed channel . . . 40
4.1.1 Gate capacitance . . . 40
4.1.2 Drift current . . . 42
4.1.3 Convection current . . . 42
4.1.4 Diffusion current . . . 43
4.1.5 Carrier temperature . . . 43
4.2 Verification of the model implementation . . . 44
4.2.1 Charge control model . . . 44
4.2.2 Parameter extraction from DC resistance . . . 46
4.2.3 Comparison of analytic calculations and numerical circuit sim- ulations . . . 47
5 Simulations 52 5.1 Detection sensitivity in the resonant plasmonic mixing regime . . . . 52
5.2 Circuit simulations with ungated access regions . . . 56
5.2.1 Power distribution . . . 56
5.2.2 Influence on plasmonic mixing efficiency . . . 60
6 TeraFET characterization 62 6.1 TeraFET figures of merit . . . 63
6.1.1 NEP and thermal noise of zero-biased TeraFETs . . . 64
6.1.2 Optical versus electrical responsivity and NEP . . . 66
6.2 Experimental setups . . . 68
6.2.1 DC measurements . . . 68
6.2.2 THz sources . . . 69
6.2.3 Detector module . . . 73
6.3 AlGaN/GaN TeraFETs . . . 74
6.3.1 First detector generation, variation of gate width . . . 75
6.3.2 Improved detectors, bow-tie and log-spiral design . . . 78
6.3.3 Highly sensitive broadband AlGaN/GaN TeraFETs . . . 84
6.4 Graphene TeraFETs . . . 88
6.4.1 Differences for graphene TeraFET model . . . 89
6.4.2 Graphene DC parameter extraction and modifications of trans- port model . . . 90
6.4.3 THz detection with graphene TeraFETs . . . 95
7 Thermoelectrics in TeraFETs 98 7.1 Origin of thermoelectric signals in TeraFETs . . . 100
7.1.1 Derivation of Seebeck coefficient from transport equations . . 100
7.1.2 Direction of thermoelectric signals . . . 102
7.2 Experimental evidences . . . 104
7.2.1 AlGaN/GaN TeraFETs . . . 105
7.2.2 Graphene TeraFETs . . . 109
8 Summary and Outlook 115
Appendices
A Plasma wave mixing model 121
A.1 Plasma velocity and plasma wavevector . . . 121
A.2 THz response . . . 123
A.2.1 Quasi-static TeraFET response . . . 124
B Derivation of transport models from the Boltzmann transport equation 126 B.1 Method of moments . . . 126
B.1.1 Charge carrier density . . . 127
B.1.2 Carrier momentum density and drift velocity . . . 128
B.1.3 Energy density . . . 128
B.2 Transport models . . . 129
B.2.1 Charge continuity equation . . . 129
B.2.2 Drift-diffusion model . . . 130
B.2.3 Hydrodynamic transport model . . . 131
B.3 Diffusion constant and Seebeck coefficient . . . 133
C Graphene TeraFET model 135 C.1 Cyclotron mass . . . 135
List of own publications 136
Bibliography 138
Zusammenfassung 149
Chapter 1 Introduction
Field-effect transistors (FETs) can act as rectifying detectors for incident electro- magnetic radiation, in particular at high frequencies. Depending on the regime of radiation frequencies and specific device design, several physical detection mechanism can take place in the FET’s channel. In this thesis, two main mechanisms of rectifi- cation in FET-based THz detectors are discussed - non-linear plasma wave mixing as well as diffusive charge carrier transport, mainly, the hot electron thermoelectric effect. A respective detector model based on a macroscopic hydrodynamic model to describe the intrinsic charge carrier transport in the FET channel was developed and is presented. The model, which extends the commonly employed fluid-dynamic description of plasma wave mixing, was translated into a transmission line equivalent circuit picture and implemented in a numerical circuit simulation software environ- ment allowing to perform numerical modeling and simulation of the photoresponse of FET-based THz detectors. Based on these considerations, a number of broad- band THz detectors were implemented in different materials, namely AlGaN/GaN HEMTs and single-layer CVD-grown graphene FETs. Results of the respective THz characterization experiments are presented and compared to simulations with the numerical model. In both materials, record sensitivities for FET-based THz detectors were obtained .
1.1 Detection mechanisms in TeraFETs
Dyakonov and Shur in the early to mid-1990s discussed theoretically that the two- dimensional electron gas (2DEG) in a semiconductor FET under appropriate radiation coupling boundary conditions can act as a cavity for charge density (plasma) waves induced by an incoming high frequency (HF) signal [18]. The detection mechanism then relies on non-linear self-mixing properties of the plasma waves launched into the transistor channel from either the drain or source terminals. It can be described with the help of a fluid-dynamic picture of charge carrier transport. This led to the term plasma wave detectors or plasmonic detectors in the literature. From a more general point of view, depending on the regime of applied frequencies, the principle manifests as classical resistive mixing (for low frequencies) or distributed resistive mixing (for high frequencies). For further increasing frequencies towards the THz spectral region,
significant enhancement of distributed resistive mixing by self-mixing properties of induced plasma waves can occur [19]. It was in particular shown that for typical semiconductor structures and materials, the frequencies for an efficient enhancement of detection can lie in the THz region. The plasma wave detection mechanism and the description by fluid dynamics has been extensively discussed over the past two decades and numerous detector implementations have been demonstrated to yield sensitivities well comparable to competing detector technologies for the THz region [20]–[22].
Another fundamental physical mechanism which can take place in TeraFETs is the generation of diffusion currents. These currents can be the result of spatial gradients in carrier density or carrier temperature in the channel induced by the incoming radiation. In the latter case, the induced diffusion signal is thermoelectric[23].
Employing FETs as efficient plasmonic detectors for electromagnetic radiation requires the implementation of asymmetric boundary conditions to ensures that the plasma waves are launched into the channel from one terminal only and no counteracting signal is created by a reverse wave from the other terminal [2], [24], [25]. As a direct consequence of this built-in asymmetry, a non-uniform distribution of incident radiation power builds up over the transistor channel. When the carrier ensemble is sufficiently decoupled from the crystal lattice, charge carriers can be locally heated on one end of the channel, while remaining at equilibrium temperature at the other end.
This situation is achieved, e.g., when energy is distributed within the carrier electronic system by multiple carrier-carrier collisions rather than by phonon scattering to the lattice. A spatial gradient in carrier temperature is the result and hot carriers will then tend to diffuse from the heated to the cold end of the channel and a significant hot-electron thermoelectric current can be observed [23], [26]–[29]. Such signals in TeraFETs have already been observed in a small number of experimental studies [5], [11], [12], [30]–[32] and were to some extent addressed in theoretical discussions [33], [34]. However, a detailed modeling of the effect in the scope of THz detection with TeraFETs is not yet available in the literature.
Following the same principle as discussed above, the local modulation of carrier density can lead to a signal contribution due to diffusion of carriers along the density gradient from channel regions of higher to lower density. We find, however, that this effect plays only a minor role in TeraFETs and it is not treated in further detail in this thesis. Note that in a thermodynamic picture, the product of carrier density and temperature can be associated with the hydrodynamic pressure via the relation P =nkBTC well known from ideal gas theory. Common state-of-the-art fluid-dynamic TeraFET models found in the literature omit pressure terms and therefore do not consider diffusive current contributions to the TeraFET detection signals [18], [35], [36].
There are mainly two reasons why thermoelectric detection has so-far mostly been omitted in discussions on plasmonic detection models for TeraFETs in the literature. First, the additional contributions are small compared to the plasma wave rectification signals in TeraFET fabricated in most conventional semiconductor materials. The most efficient detectors have been fabricated in Si-CMOS [24], [37], which is foremost due to the advanced maturity of the technology[21]. No signal contributions which could be related to diffusion of charge carriers had become
1.2 Hydrodynamic transport model and circuit implementation
evident in the multitude of THz detection experiments. However, during the course of investigation of TeraFETs in new material systems, in AlGaN/GaN HEMTs and particularly in graphene FETs, could the effect be observed in THz detection experiments to a significant level at room temperature.
A second aspect is that thermoelectric signals in conventional semiconductor TeraFETs appear most prominently at detector operation points (in terms of applied gate bias voltage) which are quite far away from operation points of best plasmonic detection sensitivity - usually close to the transistor’s threshold voltage. At gate biases where the plasmonic signals are dominating, thermoelectric contributions to the signal can easily be masked by the plasma wave mixing signal. However, since the plasmonic detector response falls off to zero rapidly with increasing gate voltage above threshold,1 the diffusive currents can appear as an isolated detection signal at high gate bias TeraFET operation.
The thermoelectric signals in the AlGaN/GaN TeraFETs investigated over the course of this work were about one order of magnitude lower than the plasma wave mixing signals. In the graphene TeraFETs, on the other hand, the situation is fundamentally different. A highly efficient decoupling of the heated carrier ensemble from the crystal lattice [27]–[29], [38] leads to pronounced thermoelectric signals on the same order as the maximum plasmonic response. Even an enhancement of the plasmonic detection is obtained. This promising observation motivates further investigation of the thermoelectric effect in TeraFETs and could in the future eventually lead to a new type of THz detector exploiting mainly the thermoelectric detection principle.
1.2 Hydrodynamic transport model and circuit implementation
The plasma wave detection mechanism in TeraFETs can be described by a set of hydrodynamic transport equations[18], [39], [40]. The formation of gated plasmons in the transistor channel is associated with a simultaneous modulation of charge carrier density and velocity. This collective excitation of charge carrier density waves is the basis for non-linear self-mixing and thereby rectification of HF radiation in an FET channel. The governing hyperbolic coupled differential equations can in principle be solved by, e.g., FDTD approaches. However, a “real” detector implemented in a physical device - as opposed to a theoretical intrinsic FET channel - in general contains a number of extrinsic device elements. Such elements may be, e.g., ungated (access) regions of the channel and other parasitic resistances, intentionally or inadvertently implemented parasitic capacitances, integrated radiation coupling antennas. The sum of all these external components determines the boundary conditions and efficiency of power transfer to the rectifying intrinsic detector element. While common detector models in the literature rely on precisely defined boundary conditions, for
1In the context of this thesis, in the case of negative threshold or Dirac voltages, the termshigh andincreasinggate bias voltage refer to “less negative” values. This is in particular important for the normally-open AlGaN/GaN HEMTs.
which analytic solutions to the governing model equations for the intrinsic detection mechanism exist, such solutions are in general not available in a real device situation.
The hydrodynamic model equations only describe the intrinsic processes in the detector’s “active” gated channel region. Although the approximate behavior of real devices can in some cases be modeled remarkably well by analysis of the hydrodynamic equations alone, it is apparent that in order to design or model a real device, the HF influence of extrinsic components must be incorporated in a realistic device model.
In particular, the power transfer from the integrated antennas to the active device regions must be considered. It was shown before that second order analysis of the transport equations can yield an analytical expression for the intrinsic TeraFET’s photoresponse given as the product of a frequency independent quasi-static term and a frequency-dependent factor accounting for plasmonic enhancement and its form depending on the specific boundary conditions of radiation coupling [18], [41], [42]. It was then proposed that approximate modeling of the THz response of a real detector can be achieved by additional multiplication of several frequency-dependent factors considering power transfer to the active transistor region depending on the HF behavior of the complex impedances of intrinsic and extrinsic device components [42][43, ch. 3]. To achieve optimal transfer of incident radiation power to the rectifying transistor region, impedance matching of extrinsic and intrinsic device elements in due consideration of the above influences should be aimed for.
In this thesis we present a somewhat different approach, namely the implementa- tion of the hydrodynamic transport equation in a circuit simulation environment.
In this way, extrinsic device components can be included in a numerical HF de- vice simulation, even when simple analytic expressions for the above HF factors or specific boundary conditions do not exist. It can be easily shown that the set of differential equations of the hydrodynamic transport model can be translated into a mathematically similar set of differential voltage and current equations known from transmission line theory of electrodynamics [44]. It was therefore suggested to model the transistor channel of a TeraFET as a distributed RCL transmission line [45]–[50]. The implementation in a circuit simulation environment then allows to include extrinsic detector components such as parasitic resistances and capacitances, ungated channel regions, integrated antenna structures, and a specific radiation cou- pling situation can be modeled to yield a realistic device simulation for real TeraFET detectors. In the course of this work, a transmission line equivalent circuit model was implemented in a commercial circuit simulation software.2 The implementation of the hydrodynamic transport model - in particular including thermoelectric signal contributions - is presented in this thesis. Simulation results for intrinsic detection are cross-verified with an analytic device model and full device simulations are compared to THz detection measurements.
2Keysight Advanced Design System (ADS) [51]
1.3 Fabrication and experimental characterization of TeraFETs
1.3 Fabrication and experimental characterization of TeraFETs
In the course of the presented work, highly sensitive THz detectors employing FETs with integrated antennas - we term these devices TeraFETs3 - were designed and fabricated in different materials, namely, AlGaN/GaN HEMTs and CVD-grown single-layer graphene FETs. The integrated planar antenna structures were laid out for broadband detection mainly operating in the region between 0.1 and 1.2 THz.
Careful device design considering optimized power coupling, electrical stabilization of the detector environment, reduction of parasitic elements by improvements in device fabrication technology, as well as thorough experimental characterization led to record measured sensitivities for TeraFETs in the respective materials, in particular, with respect to broadband detection. The devices were fabricated in successful collab- orations with Ferdinand-Braun-Institut, Leibniz-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany (AlGaN/GaN HEMTs), and with Chalmers University of
Technology, Göteborg, Sweden (graphene FETs).
Based on experimental observations emerging in the course of the THz characteri- zation of the above TeraFETs, we will show in this thesis that the implementation of optimal boundary conditions for an efficient plasma wave-based non-linear rectifica- tion - i.e., asymmetric coupling of the incident radiation to the transistor terminals - in turn gives rise to significant thermoelectric signals due to local heating of charge carriers in the FET channel. With the help of the implemented hydrodynamic detector model including diffusion terms, the measurement results are discussed and reproduced qualitatively, which suggests that an extension of state-of-the-art TeraFET models commonly found in the literature is required to account for this additional detection mechanism in TeraFETs.
Some of the presented results from THz measurements have in parts been pub- lished prior to the writing of this thesis. The respective references are Refs. [2], [3], [9]–[11], [13], [15] for the AlGaN/GaN HEMTs and Refs. [5], [10], [12] for the graphene FETs.
1.4 Structure of this thesis
This thesis is structured as follows. Chapter 2 presents the basic theory and gives some discussion of the intrinsic plasma wave-based detection mechanism. The fluid- dynamic transport description as employed by Dyakonov and Shur [18] is presented and some important equations for fundamental quantities, such as channel impedance and plasmonic efficiency factor, are derived. The requirement and consequences of asymmetric boundary conditions are discussed and motivate the extension of the detector model by diffusive transport contributions. An transmission line equivalent circuit description of the model equations is introduced.
3We introduce the term TeraFETs to emphasize the distinction between the intrinsic detector element, i.e., an FET channel, and real physical detectors designed for THz detection and consisting of an antenna-integrated FET including extrinsic detector elements.
Chapter 3 presents the formulation of a comprehensive hydrodynamic transport model from basic transport theory. Model equations for charge carrier density, current density and energy density are obtained from the fundamental Boltzmann transport equation by application of the method of moments. A final comparison of the fluid- dynamic transport description from Ref. [18] with the derived hydrodynamic model equations confirms, that the latter extends the transport description by diffusive terms.
In Chapter 4, the implementation of the hydrodynamic transport model in a transmission line equivalent circuit picture in a commercial circuit simulation soft- ware [51] is presented and some details of the implementation are addressed. The software allows numerical simulation of DC device parameters and THz rectification.
Simulation results are verified by comparison with results obtained from analytic expressions derived in Chapter 2. Both the numerical simulations with the circuit model as well as the analytic calculation results were based on realistic device param- eters extracted from DC characteristics of fabricated devices. A fitting procedure for parameter extraction from DC measured drain-source resistances of the TeraFETs is presented considering the example of a GaN-based detector.
Chapter 5 presents simulation results obtained with the implemented device model, in particular, with respect to the intrinsic plasmonic enhancement factors and the inclusion of ungated channel regions in the detector’s equivalent circuit. The concept of resonant plasmonic enhancement is investigated in some detail. Power distribution over the transistor elements is investigated and the influence of power loss to ungated channel regions on the efficiency of plasmonic rectification is discussed.
Experimental results of THz characterization experiments with the fabricated AlGaN/GaN and graphene TeraFETs are presented in Chapter 6. Important figures of merit for the evaluation of THz detectors are introduced and the employed experimental setup is outlined. Three generations of AlGaN/GaN TeraFETs are presented, which on the basis of extensive experimental and theoretical studies and constant design optimization eventually yielded highly sensitive broadband detectors in the frequency region of 0.2 to 1.2 THz. The detectors exhibited record performance for GaN-based TeraFETs and the respective measurement results are presented. The achieved results are currently being prepared for an invited publication [1]. CVD-grown Graphene TeraFETs were produced roughly based on the design of the AlGaN/GaN TeraFETs. The detectors also yielded record sensitivities for graphene TeraFETs at 590 GHz[5]. The measurements are presented and necessary modifications to the model implementation for graphene FETs are discussed.
Significant thermoelectric contributions to the THz detection signals were observed in both materials. These experimental findings are summarized in Chapter 7 together with some further discussion. Comparison of the measurement results with device simulations show that implemented model including thermoelectric contributions can account for the observed signals on a good qualitative level.
Chapter 2
Plasma wave mixing model
In this chapter we present the underlying theoretical model which is employed to describe the plasma wave-based rectification of THz radiation in FETs. The model is presented following the original formulation of Dyakonov and Shur [18].
We investigate the detection regimes of classical resistive, distributed resistive and plasmonic mixing [19] and discuss the requirement and consequences of asymmetric boundary conditions of radiation coupling. Finally, we present a transmission line equivalent formulation of the detector model, whose implementation in a circuit model-based device simulation is shown in Chapter 4.
2.1 Foundation of the concept of plasma wave mixing
It has long been known from basic electronic transport theory in semiconductors that charge carrier dynamics can be well described by fluid dynamic transport models. In an early work, Bloch introduced the concept of using a hydrodynamic treatment based on the Thomas-Fermi atomic model to describe the dynamics of charge carriers as an electron (Fermi-)gas in a semiconductor [52]. He showed that the governing equations of motion are indeed Euler’s equation of motion together with the fundamental continuity equation. This system of equations had originally been developed in the 18th century for the description of fluid dynamical processes of gases and liquids [53]. Hydrodynamic and related macroscopic transport models today find wide application in the analysis of semiconductor devices. The basis of these models is to describe carrier transport by averaged macroscopic quantities of the system of charge carriers instead of the individual particles, e.g., the carrier density, current density, and energy density [54], [55].
Due to the identity of description of fluids and charge carriers in semiconductors, many wave-like phenomena known from fluid dynamics have corresponding analogies in carrier transport problems. In particular in FETs, the excitation of propagating collective charge density oscillations in the 2DEG of a semiconductor, so-called plasma waves, is possible and resembles to a large extent the physics of shallow water waves [39]. Originally, the general concept of plasma waves has been developed for three-dimensional systems by Tonks and Langmuir in the late 1920s [56] and was
discussed for bulk metals in great detail by Raimes in Ref. [57] and some references given therein. The electron ensemble in a semiconductor in equilibrium constitutes a uniform distribution of charge carriers in the periodic background potential of positively charged crystal ions. When an external field is applied for a short time, the electron gas is shifted relative to the background and, in order to restore the original distribution, a polarization field is induced, which drives the electrons back to their initial position. Because of their finite inertia, charges can overshoot this initial position and when the external force is periodic, collective longitudinal oscillations of charge carriers around their initial position can evolve.
An according formalism for lower dimensional systems, e.g., in semiconductor inversion layers, was first developed in the late 1960s by Stern [58] and discussed in further detail by Chaplik [59], [60] and Ando [61]. It was found that a linear dispersion relation can exist in these semiconductor structures. In 1977, absorption of infrared radiation in silicon inversion layers was first experimentally observed by Allen [62] and later discussed in great detail by Chaplik [60]. In these original works, a necessary spatial modulation of the electric field was realized by metallic grating structures to allow matching of the plasmon wavevectors to the incident radiation.
In the early to mid-1990s, Dyakonov and Shur showed in a number of pioneering works [18], [39], [40] that the same mechanism can take place in the 2DEG of a FET. They employed a reduced4 fluid dynamic transport model for shallow water waves to describe the charge carrier transport by propagating plasma waves in the channel [39] and demonstrated theoretically that non-linear self-mixing properties of the waves can be used for an efficient detection of radiation in the THz region [18].
The concept was divided into two main regimes of detector operation.
- Resonant detection: When the waves launched from one side of the transistor are reflected off the second terminal, a standing wave pattern inside the channel acting as a resonant cavity can develop, and a strong enhancement of the detection mechanism by some orders of magnitude was expected [18]. Very few experimental investigations of this proposed phenomenon are at all available and it seems evident that an appropriate implementation of a real device is difficult to achieve with common semiconductor technologies. At least for conventional materials, extremely small device dimensions, cryogenic cooling of the detector [63]–[65] or drawing of an additional drain-source current is required [66]. The theoretical predictions from Ref. [18] have led to a questionable premise in the search for resonant plasmons in FETs. Although strong resonant enhancement of the mixing principle itself can exists, we will discuss in Chapter 5 that a pronounced enhancement of detection sensitivity cannot be expected for resonant plasmonic detection due to resonant power coupling at these frequencies.
- Non-resonant/ broadband detection: When the transistor channel is much longer than the typical decay length of the plasma waves, the detector operates in the so-called non-resonant detection regime [18]. The non-linear mixing happens on
4It will be worked out in Chapter 3 that the model is a “reduced” one in the sense that pressure terms and the effect of local heating of charge carriers are not considered in the formalism.
2.2 One-dimensional treatment of charge carrier transport
a rather short range in the channel and the remaining parts of the channel act as parasitic resistance and AC shunt capacitances for the incoming radiation.
Naturally, this mechanism is not restricted to an exact resonance frequency and allows broadband detection of electromagnetic radiation [13], [24], [48], [67]–[70]. Also, no cooling is needed to achieve nearly state-of-the-art detector sensitivities for the THz region and TeraFETs can be efficiently operated at room temperature[20], [43], [71].
While the fundamental mechanism of the plasmonic detection process is largely understood, modeling and realization of an actual device remains a challenging task.
Parasitic effects of the surrounding of the intrinsic gated transistor region as well as integration of an antenna structure for radiation coupling and efficient implementation of boundary conditions plays a crucial role for the TeraFET’s efficiency. These aspects can to some extent be incorporated in an analytical device model by simple considerations of an equivalent circuit model of the detector. However, the analytical model, which will be presented in this chapter, relies on assumptions of exactly matched boundary conditions for the coupling of the incoming radiation, otherwise, the underlying equations may lose their validity. These boundary conditions are hardly ever perfectly met in real detectors, in particular when broadband operation is desired. This is one of the main reasons why implementation of the model in a circuit simulator can be advantageous to model non-ideal boundary conditions in a device simulation.
This following sections reviews the use of a reduced fluid dynamic model for the description of charge carrier transport in an FET’s 2DEG. We will then show in Chapter 3 that this reduced form of Euler’s equation together with the charge continuity equation is contained within a full hydrodynamic transport description [54, p. 201], which can be derived from the fundamental Boltzmann transport equation, and which contains additional density diffusive and thermoelectric current densities.
2.2 One-dimensional treatment of charge carrier transport
Assume a simplified transistor layout as depicted in Fig. 2.1. A 2DEG can form in an inversion layer at the boundary of the bulk semiconductor and the gate dielectric barrier. Application of a voltage to the gate metal contact controls the electrostatic potential in the inversion layer and thereby the charge carrier density in the 2DEG.
While a real transistor is (except for intentionally designed low-dimensional structures) a three dimensional structure, it can be shown that the charge carrier dynamics in the context of FETs employed as high frequency detectors can be described by a one dimensional transport model. First, the device is assumed to be fully symmetric in the y-direction with respect to the x-z-plane. Second, it is assumed that the longitudinal electric field in the channel varies only gradually compared to the local gate-to-channel potential along the along the y-direction. This so-called gradual channel approximation is valid for non-saturation operation oft the transistor and in general for small gate-to-channel separations d, which is the case for typical semiconductor FETs where d is on the order of a few up to a few tens of nanometers.
bulk semiconductor (εS) barrier (ε)
2-DEG
Drain Source x Gate
y z
W
Lug Lg Lug
d
dS
Fig. 2.1: Simplified schematic of a field-effect transistor (FET) and labeling of charac- teristic device geometrical parameters. Description of carrier transport is reduced to a one-dimensional treatment along the x-direction.
With these two assumptions, the carrier transport in the device can be described by a one-dimensional model along the x-direction. Note that this assumption implies that there exist no significant oblique contributions to the plasma waves in the channel, in particular no turbulence effects. It has been discussed to some extent that this may not be the case for high field or high drain current situations [33], [72].
When the carrier transport in a semiconductor device is non-ballistic, i.e., when carriers experience a multitude of collisions during transit and the mean free path between two consecutive scattering events is short compared to the device dimension, fluid dynamic transport models apply [54]. In particular for most semiconductor FETs, a high density of carriers (∼1012cm−1) in the 2DEG leads to frequent electron- electron (e-e) collisions and to a distribution of excess energy in the carrier ensemble, which justifies the application of a hydrodynamic description [39]. Dyakonov and Shur investigated the possibility of frequency conversion in such electronic systems by applying analogies of instability mechanism known from the behavior of shallow water waves or sound waves in organ pipes [18], [39], [40], [73].
Fluid dynamic motion can be described by Euler’s equation of motion together with the equation for charge continuity. When pressure terms are neglected and a phenomenological friction term is added, the transport model can be formulated as5, 6
∂tv+v∂xv− q
m∂xφ+ v
τp = 0 (2.1)
∂tn+∂x(nv) = 0 (2.2)
In the model,v is the average carrier drift velocity, nthe carrier density, and φ is the gate-to-channel potential. The constants q,m, and τp are the elementary charge, the effective carrier mass, and the momentum scattering time, respectively. When the potentialφ is modulated by an oscillating external signal, collective charge density oscillations will be induced in the channel, which, by non-linear self-mixing, can
5In this text, the abbreviated notations ∂t ≡ ∂/∂t and ∂x ≡ ∂/∂x are used denoting partial derivatives with respect to timet andx-coordinate, respectively
6The model is formulated in terms of carrier density and drift velocityn andv, respectively. It can be rewritten in terms of the current densityj =−qnv and then be directly implemented in a circuit simulator (see Chapters 3 and 4 ).
2.3 Intrinsic channel impedance and THz response
rectify the incoming signal and induce a measurable DC current (or voltage) response in the device.
2.3 Intrinsic channel impedance and THz response
In this section, some important physical quantities for the discussion of plasma wave mixing in TeraFETs are derived. The discussion is limited to the intrinsic gated region of the transistor channel, since influences of extrinsic detector elements cannot be easily treated in analytic form. While extension of a quasi-static rectification picture by HF factors to account for changes of HF device impedances at high frequencies, matching of integrated antennas, etc. has been proposed before [42], calculation of channel impedance, carrier density and velocity as well as plasmonic enhancement of resistive mixing rely on defined boundary conditions to yield analytic expressions for these quantities (see below). For non-ideal coupling situations, analytic solutions to the transport equations Eqs. (2.1) and (2.2) in general do not exist. Furthermore, power distribution over the device plays a crucial role for the efficiency of the rectification mechanism and must be considered in device design and modeling and will be investigated in Chapter 5. Nevertheless, comparison of analytically derived results for the intrinsic detector element served as a benchmark for the verification of the model implementation presented in Chapter 4. The discussion in this and the following sections are mainly based on Refs. [18], [19], [42], [43]. Some details on the underlying calculations are given in Appendix A.
2.3.1 Channel impedance and carrier velocity
Analysis of plasma wave mixing in TeraFETs described by the fluid-dynamic model is commonly performed by harmonic expansion of the applied oscillating signal and related quantities in terms of the oscillation angular frequency ω. An analysis to the first order in ω of the transport equations Eqs. (2.1) and (2.2) leads to a wave-equation
k2φ1+∂x2φ1 = 0 (2.3)
for the oscillating component φ1 ∝ei(ωt−kx) of the channel potential.7 The solutions are propagating charge density (plasma) waves with angular frequency ω, plasma velocity
s =
sq mn0
∂φ
∂n (2.4)
and complex wavector
k = ω s
s
1− i
ωτp (2.5)
Eq. (2.5) shows that the gated plasma waves obey a linear dispersion relation for ωτp >>1, the so-called plasmonic mixing regime (see Section 2.4.3). The channel
7The subscript denotes the order in frequencyω in the harmonic expansion.
G
S D
V cos(ωt) VG
a
G
S D
V cos(ωt) VG a
V / Idet det
Drain coupling Gate-source coupling
V / Idet det
Fig. 2.2: Equivalent circuit for drain coupling (left) and gate-source coupling (right) boundary conditions.
potential is related to the local gate voltage-controlled charge carrier density in the channel by Poisson’s equation and can be expressed for the first order in ω in a long-wavelength approximation (see Appendix A.1) as [42]
φ1(x, t) =−n1(x, t)∂φ
∂n (2.6)
Asymmetric boundary conditions are required for an efficient detection of incoming radiation (for more details, see Section 2.5 below). Although plasma oscillations can still form in fully symmetrically excited FET channels, no net measurable voltage (or current) response to the incident radiation can build up. Full asymmetry is achieved, e.g., when one end of the channel is subjected to the full amplitude of the applied excitation signal Vacos(ωt) while the other end of the channel is pinned to zero amplitude
φ1(x= 0) = 0
φ1(x=Lg) = Va (2.7)
The picture of charge density waves launched into the transistor channel by the incoming radiation from one end, in this case the drain side atx=Lg, is established.
This coupling scheme is therefore referred to asdrain coupling. A schematic illustration of the situation is shown in Fig. 2.2 on the left. With given boundary conditions, the solutions to the wave equation Eq. (2.3) are uniquely determined. The amplitude of the modulation of the channel potential is then directly related to the first order gate-to-channel voltage amplitudeV1 and can with respect to Eq. (2.7) be calculated as [42]
V1(x) =−Va sin(kx)
sin(kLg) (2.8)
The velocity of charge carriers is modulated by the external oscillation signal and becomes
v1(x) = Va qω iks2m
cos(kx) sin(kLg)
=Va ω ikn0
∂n
∂φ
cos(kx) sin(kLg)
(2.9)
From the carrier velocity, the current density can be calculated via the well-known
2.3 Intrinsic channel impedance and THz response
relation j1 =−qn0v1 and the AC impedance of the channel is simply given as the ratio of voltage and total current I =j1W at the drain8
Zg = V1(x) j1(x)W
x=L
g
= iks2m
ωq2n0W tan(kLg)
= ikn0 ωW
∂n
∂φtan(kLg)
(2.10)
Figure 2.2 on the right shows coupling conditions as proposed in the original works of Dyakonov and Shur [18]. The incoming radiation is coupled to the FET between the gate and source while the detected signal is again read out at the drain end of the channel. These boundary conditions are commonly referred to as gate-source coupling and are defined as
φ1(x= 0) =Va
v1(x=Lg) = 0 (2.11)
with full oscillation of the gate-to-channel potential at the source side and vanishing first order velocity component at the drain end of the channel. Following a similar derivation as above, the gate-to-channel voltage and carrier velocity are calculated as
V1,GS(x) =−Vacos(k(Lg −x))
cos(kLg) (2.12)
v1,GS(x) =Va qω iks2m
sin(k(Lg−x))
sin(kLg) (2.13)
and the channel impedance is found to be
Zg,GS= V1,GS(x) j1,GS(x)W
x=0
= iks2m
ωq2n0W cot(kLg)
= ikn0 ωW
∂n
∂φcot(kLg)
(2.14)
The channel impedance is a crucial design parameter for TeraFET detectors.
In a real device with attached antenna and possible parasitic circuit elements, the amount of power of the incoming radiation which is available for the actual plasmonic detection mechanism is distributed over the device elements depending on their relative high frequency impedances according to a simple voltage divider picture (see Chapter 5). In particular, significant amounts of the incoming power can be “lost”
to ungated access regions of the transistor.
8Note again that the expressions for gate-to-channel voltage, carrier velocity and channel impedance apply only for the case of AC source-gate shunting and full oscillation at the drain side of the channel. They take slightly different forms for other coupling schemes (see Appendix A.2.1). For a more detailed overview see, e.g., Ref. [43, ch. 3]
2.3.2 THz response
Second order analysis of the fluid-dynamic model yields the DC induced detector response to an applied high frequency signal with angular frequencyω. Non-linear mixing of induced plasma waves in the channel leads to rectification of the signal, which is the basis for the plasmonic mixing principle. The FET’s DC voltage response is found by integration of the differential transport equations Eqs. (2.1) and (2.2) over the active gated channel length Lg retaining all oscillatory terms up to the second order inω, which produce a stationary response to the applied external signal.
The induced detector signal Vdet can then be derived as (see Appendix A.2) Vdet=V(x=Lg)−V(x= 0) = q
m Va2
4s2f(ω). (2.15) the product of a quasi-stationary response and a frequency-dependentenhancement orefficiency factor f(ω), whose exact form again depends on the applied boundary conditions. Note that the detection signal can also be read out as current signal where the relation
Idet = Vdet
RDC (2.16)
with the FET’s DC drain-source resistance RDC holds.
For the drain coupling, AC gate-source shunting boundary conditions defined in Eq. (2.7) the enhancement factor takes the form [42], [43]
fD(ω) = 1 +βsinh2(kiLg)−sin2(krLg) cosh2(kiLg)−cos2(krLg)
β = 2
r
1 +ωτ1p2
(2.17)
where kr and ki are the real and imaginary parts of the wavevector k= kr−iki from Eq. (2.5). For coupling of the high frequency signal between gate and source and open drain terminal, the efficiency factor is calculated as [18], [43]
fGS(ω) = 1 +β− 1 +βcos (2krLg)
sinh2(kiLg) + cos2(krL) (2.18) A simplified factor can be defined for the broadband, long-channel case where no significant reflections at the channel ends occur. In this regime, the oscillatory terms in both equations Eqs. (2.17) or (2.18) reduce to unity for kLg 1 and
fs(ω) = 1 +β = 1 + 2
r
1 +ωτ1p2 (2.19)
The simplified efficiency factor asymptotically approaches a maximum value of three for largeω [18], [43]. Further discussion of the enhancement factor is given in Section 2.4.3.
The quasi-static voltage response can be shown to take the same form as the
2.3 Intrinsic channel impedance and THz response
pre-factor in Eq. (2.15) (see Appendix A.2.1) and with the definition of plasma velocity Eq. (2.4) we find
VQS= q m
Va2 4s2
= Va2 4
1 n0
∂n
∂φ
= Va2 4
1 σ
∂σ
∂φ
(2.20)
where we introduced the DC channel conductivity
σ=qn0µ (2.21)
This remarkable result implies that the quasi-static voltage response of a TeraFET can be predicted from its DC channel conductance G(VG) = σW/Lg, which is accessible via conventional DC I-V measurements. However, a few remarks should be made concerning this observation.
First, a similar result was obtained by neglecting all but the field-dependent term in Eq. (2.1) corresponding to j = −σ∂xφ [74]. The approach reveals that in the quasi-static analysis, plasma wave phenomena are not considered and the prediction of the detector response after Eq.(2.20) is based purely on resistive mixing (see below).
While this can be justified for low-frequency application, the inconsiderate use of the formula in the discussion of THz detection experiments is highly questionable though unfortunately often found in TeraFET literature. Plasmonic enhancement of the quasi-static response is accounted for in the full detector response Eq. (2.15) by the frequency-dependent enhancement factor f(ω) (for further discussion, see Section 2.4.3).
Second, besides plasmonic enhancement, the THz response of a real TeraFET is strongly influenced by the HF behavior of the channel impedance Zg (Eq. (2.10)), frequency-dependent impedance matching of integrated antenna structures, HF power distribution over the gated and ungated channel regions, and influences of other parasitic device elements. It can therefore not be expected that a simple analysis of DC device parameters is sufficient to predict HF detection characteristics. The quasi-static response must therefore be embedded in a more comprehensive formalism accounting for the above phenomena, as was suggested, e.g., in Ref. [42]. Lastly, it will be shown in this thesis that other physical mechanism of THz rectification may take place in TeraFETs, which are naturally not covered by the presented analysis of plasma wave mixing.
In conclusion, the quasi-static TeraFET response after Eq. (2.20) should be applied only as a rule-of-thumb and has shown to yield remarkably good approximations to measured THz detection signals in some situations. It must however be emphasized that this can be misleading since important HF phenomena are not considered and deviations from real THz detection signals are often substantial. Another important aspect here is that the amplitude of the oscillating signal at the gated channel region Va is commonly not an accessible parameter in real detection experiments.
In general, a normalization of equation EQ (2.20) is therefore necessary, which can mask significant deviations between measured and predicted THz responses. This underlines the need for the implementation of the intrinsic plasma wave detection
Table 2.1: Detection regimes of TeraFETs (in reference to Table 3 in [43]) Wavelength \ Frequency ωτp <1
strong damping
ωτp>1 weak damping Re{kLg}<1
short channel
Classical resistive mixing
Resonant plasmonic mixing
Re{kLg}>1 long channel
Distributed resistive mixing
Non−resonant plasmonic mixing
mechanism in a full device model accounting for extrinsic influences on TeraFET detection signals.
2.4 Detection regimes
The above plasma wave model naturally leads to various detection regimes in which the detector can be operated. Depending on the frequency of the applied radiation, the detection principle is classical resistive mixing by modulation of the channel conductance, or distributed resistive mixing orplasmonic mixing by non-linear self- mixing properties of induced charge density waves. From an equivalent circuit point of view, when the wavelength of the induced plasma oscillations is much longer than the channel length, the transistor’s 2DEG channel can be represented by a single lumped RC element and the wave picture can be neglected. On the other hand, for higher frequencies where the plasma wavelength becomes comparable to or shorter than the channel length, the lumped element picture loses its validity. The local modulation of carrier density and velocity in the channel by the applied signal become important and require a distributed element description [19]. The transistor’s gated channel region must then be treated as a transmission line composed of RLC unit cells [44], [46], [48]. As is known from basic transmission line theory (e.g., [75], [76]), a wave-based description identical to the solutions of the fluid-dynamic model Eqs.
(2.1) and (2.2) is then suitable to solve the fundamental set of differential equations for channel voltage and current – the telegrapher’s equations (see Section 2.6).
Dyakonov and Shur discussed the possibility of rectification of HF signals in FETs from a charge carrier transport point-of-view [18] rather than a transmission line approach. While their formulation of the detection model was based on fluid dynamic differential equations for carrier density n and drift velocity v, both sets of equations are equally valid for the description of the underlying physical detection processes – one stemming from a more physics-based viewpoint, the other from an electrical engineering approach.
The plasma wavevector in Eq. (2.5) is a complex quantity and hence describes propagation and attenuation of the waves. A characterization of detector operation regimes is commonly based on the capability of the induced plasma waves to propagate along the channel. Two criteria can be formulated for this purpose.
A transition point from strong to weak damping can be loosely defined at the
